Parseval formülü yardımıyla genelleşmiş fonksiyon katsayılı ikinci mertebeden fark denklemleri için ters spektral problemler

(1)

* Corresponding Author DOI: 10.37094/adyujsci.594199

Inverse Spectral Problems for Second Order Difference Equations with

Generalized Function Potentials by aid of Parseval Formula

Bayram BALA1,_*

1_{Harran University, Department of Mathematics, Şanlıurfa, Turkey.}

bbala@harran.edu.tr, ORCID: 0000-0001-6515-6039

Received: 19.07.2019 Accepted: 03.03.2020 Published: 25.06.2020

Abstract

In the present study we are investigated inverse spectral problems for spectral analysis and two spectra of matrix by using equality which is equivalance Parseval formula. The matrix is tridiagonal almost-symmetric matrix. The mean of almost-symmetric is the entries above and below the main diagonal are the same except the entries and .

Keywords: Parseval formula; Spectral analysis; Two spectra.

Parseval Formülü yardımıyla Genelleşmiş Fonksiyon Katsayılı İkinci Mertebeden Fark Denklemleri için Ters Spektral Problemler

Öz

Bu çalışmada, Parseval formülü ile eşdeğer olan eşitlik kullanılarak matrisinin spektral analizine göre ve iki spektrumuna göre ters spektral problemleri incelenmiştir. , tipinde hemen hemen simetrik üçköşegensel matristir. Hemen hemen simetriklik, ve elemanları dışında matrisin köşegeninin altında ve üstündeki elemanları eşit olmasıdır.

J J N N´ M

a

c

_M J J N N´ M

a

c

_M

(2)

Anahtar Kelimeler: Parseval formülü; Spektral analiz; İki spektrum. 1. Introduction

Consider the second order difference equation

(1) with the boundary conditions

(2)

where is column vector which is solution of the second order difference equation, is constant

𝜌!= #

1, 0 ≤ 𝑛 ≤ 𝑀

𝛼, 𝑀 < 𝑛 ≤ 𝑁 − 1 , 𝛼 ∈ ℝ

"_{− {1}, (3)}

and 𝑎_!, 𝑏_!∈ ℝ, 𝑎_!> 0 are coefficients of Eqn. (1),

(4)

Now, we can write the Eqn. (1) by definition of

is tridiagonal almost-symmetric matrix and the entries of are the coefficients of Eqn. (1).

{

}

1 1 1

,

1 1

1,

0,1,...,

1 ,

n n n n n n n n N

a y

_- _-

+

b y

+

a y

₊

=

lr

y

a

_-

=

c

_-

=

n

Î

N

-1 N

0,

y

_-

=

y

=

{ }

1 0 N n n

y

=

y

₌ -n

r

{

}

{

}

,

1,...,

2 ,

,

1,

2,...,

1 .

n n n n

c

a

n

M M

N

d

b

n

M

N

a

=

Î

+

-=

Î

+

-n

r

{

}

{

}

1 1 1 1 1 1

,

0,1,...,

,

1,

2,...,

1 .

n n n n n n n n n n n n n n

a y

b y

a y

y

n

M

c y

d y

c y

y

n

M

N

l

- - + - - +

+

=

Î

ìï

í

+

=

Î

+

-ïî

J

N N

´

J

(3)

.

So, the eigenvalue problem is equivalent problem (1)-(3) which is discrete form Sturm-Liouville problem with discontinuous coefficients

(5)

where is a piecewise function

H. Hochstadt made significant contributions to the development of the inverse problem for second order difference equations. He studied inverse problem for Jacobi matrices in [1-4]. Later, G. Guseinov has pioneered for inverse problem of infinite symmetric tridiagonal matrices. He considered different kinds of inverse spectral problems for second order difference equation; such as the inverse spectral problems of spectral analysis for infinite Jacobi matrices in [5], the inverse spectral problems for the infinite non-self adjoint Jacobi matrices from generalized spectral function in [6, 7], and the inverse spectral problems for same matrices from spectral data and two spectra in [8-12]. The inverse spectral problem for discrete form of Sturm-Liouville problem with continuous coefficients has been studied in [13] and the inverse spectral problem with spectral parameter in the initial conditions has been studied by M. Manafov in [14]. The eigenvalues and eigenfunctions and the inverse problem for Sturm-Liouville operator with discontinuous coefficients which is the same problem given by (5) are investigated by E. Akhmedova and H.

0 0 0 1 1 1 2 1 1 1 1 1 1 2 3 3 3 2 2 2 1

0

M M M M M M M M M M N N N N N N N

b

a

b

a

b

a

b

a

J

c

d

c

d

c

d

c

d

- -+ + + + - -- - --

-é

=

ë

!

"

" #

"

#

"

!

"

" #

"

#

"

!

ù

ê

ú

ê

ú

ê

ú

ê

ú

ê

ú

ê

ú

ê

ú

ê

ú

ê

ú

ê

ú

ê

ú

ê

ú

ê

ú

ê

ú

ê

ú

ê

ú

ê

_úû

Jy

=

l

y

( )

( ) ( )

[ ]

( )

,

, ,

0,

d

p x

y x

q x y x

x y x

x

a b

dx

y a

y b

lr

é

_{ù +}

₌

_Î

ê

ú

ë

û

=

( )

x

r

( )

2 2

1,

,

1. , c

a x c

x

x b

r

a

£ £

ì

=

_í

¹

< £

î

(4)

Huseynov in [15, 16], respectively. Bala et al. are studied inverse spectral problem for almost symmetric tridiagonal matrices from generalized spectral function in [17] and they examined inverse spectral problems for same matrices from spectral data and two spectra in [18]. Finite dimensional inverse problems are investigated by H. Huseynov in [19].

Also, Parseval equality of discrete Sturm-Liouville equation with periodic generalized function potentials is studied by Manafov et al. in [20]. At the same time a new approach for higher-order difference equations and eigenvalue problems is examined by Bas and Ozarslan in [21].

The goal of this article is to study inverse spectral problems of the problem (1)-(3) for spectral analysis and two spectra by using Parseval formula.

2. Direct Problem for Spectral Analysis

The matrix has number eigenvalues and number eigenvectors , which form an orthonormalized basis. Assume that the eigenvalues are real. We bring to mind the algorithm of structure for the matrix eigenvalues and eigenvectors.

Let be a solution of Eqn. (1)

(6) with initial conditions

, (7)

and the degree of polynomial is .

Lemma 1. The following equality holds:

.

Proof. See [17].

According to Lemma 1, the roots of the equation are equal the eigenvalues of , and eigenvectors corresponding eigenvalues will be

J N

l l

₁

, ,...,

₂

l

_N N 1

, ,...,

2 N

v v

v

J

( )

n

P

l

( )

{

}

1 1 1

,

0,1,...,

1 ,

n n n n n n n n

a P

_- _-

l

+

b P

l

+

a P

₊

l lr

=

P

l

n

Î

N

-( )

( )

1

0,

0

1 P

_-

l

=

P

l

=

( )

n

P

l

n

(

) ( )

0 1 1 1

( )

det

J

-

l

I

= -

1

N

a a a c

...

_{M M}₊

...

c P

_N_- _N

l

( )

n

P

l

J

,

1,

k

N

l

=

(5)

.

Assuming that , where . Thus, we have the complete

orthonormalized system of eigenvectors of the matrix . The numbers are called normalized numbers of the problem (1)-(3).

Lemma 2. Eigenvalues of matrix are different.

Proof. Because of eigenvalues are the roots of polynomial , we must show that . Firstly, take the derivative equation Eqn. (6) by , we have

(8)

Now, if the Eqn. (8) is multiplied by and the Eqn. (6) is multiplied by , the second result is substracted from the first, for we obtain

. (9)

For and summing from to , pay attention to Eqn. (7) and we have

. (10)

As a result, .

According to Lemma 2, we can assume that . The following Lemma is about Parseval equality.

Lemma 3. The expansion formula which is equivalent Parceval equality, can be written as

below: , (11)

( )

(

0

( ) ( )

,

1

,...,

1

( )

)

T k

P

k

P

k

P

N k

l

_-

l

Â

=

( )

k k k

v

l

b

Â

=

1 2

( )

0 N k j k j

P

b

-

l

=

å

J

b

_k J

,

1, 2,...,

k

N

l

=

P

_N

( )

l

( )

0

N k

P

¢

l

¹

l

( )

1 1 1

.

n n n n n n n n n n

a P

_-

¢

_-

l

+

b P

¢

l

+

a P

¢

₊

l lr

=

P

¢

l r

+

P

l

( )

n

P

l

P

_n

¢

( )

l

{

0,1,...,

1 }

n

Î

N

-( ) -( )

( ) ( )

(

)

(

( ) ( )

)

2

( )

1 1 1 1 1 n n n n n n n n n n n n

a

_-

P

¢

_-

l

P

l

-

P

¢

l

P

_-

l

-

a P

¢

l

P

₊

l

-

P

¢

₊

l

P

l

=

r

P

l

k

l l

=

n

0 N-1

P

_N

( )

l

_k

=

0 ( )

( )

2

( )

1 2

( )

1 1 0 1 M N N N k N k j j j j M

a P

_-

l

P

_-

l

P

l

-

a

P

l

= = +

¢

=

å

+

å

( )

0

N k

P

¢

l

¹

1 2

...

N

l l

<

< <

l

( ) ( )

1

,

0,

1

N m j n j mn j j

P

m n

N

h

_l

_d

b

=

-å

(6)

where is defined by

, (12)

and is the Kronecker delta.

For in the Eqn. (11) and from conditions (7) we obtain following equality

. (13)

Thus, we get eigenvalues and eigenvectors corresponding . So, we can say that the direct spectral problem of spectral analysis is solved.

Now let's try to answer the following question:

If we know eigenvalues and eigenvectors of matrix , is it possible to reconstruct the matrix by using the following formula

,

where scalar product.

It is clear that eigenvalues of is not sufficient for reconstruct matrix . On account of this we must have some more information about eigenvectors.

Definition 4. The collection of quantitites are called spectral data for the matrix

Additional we will need the presentation of entries of the matrix by the polynomial . For the Eqn. (6) is multiplied by , then summing by from to and using Lemma 3, we have

h

1,

or

, or

m

n M

m

n M

h

a

£

ì

= í

_>

î

mn

d

0 n m= = 1

1

N j=

b

j

=

å

{ }

N₁ k k

l

₌

v

_j

,

j

=

1, 2,...,

N

{ }

N₁ k k

l

₌

{ }

N₁ k k

l

₌

{ }

N₁ k k

v

₌ J J

(

)

2

(

)

1 , , 0, 1 N k k k k Ju

l

u v v u l N = =

å

Î

-( )

1 0

,

N _{j j} j

u v

-

u v

=

å

J J

{

l b

_k

,

_k

}

. J J

( )

n

P

l

l l

=

_j _m

( )

_j j

P

h

_l

b

j

1

N

(7)

where is defined by (12). It is clear that for . Then, we can write these equalities as below:

(14)

(15)

(16)

(17)

(18)

3. Inverse Problem of Spectral Analysis The inverse problem of spectral analysis is reconstruct matrix by using the collection quantities . Theorem 5. Let an arbitrary collection of matrix is . In order for this collection to be spectral data for some matrix which have form , it is necessary and sufficient that the following conditions are satisfied: (i) , (ii) , (iii) .

( ) ( )

2 1 1

,

0, N 2 M,

N j n n j n j j j

a

h l

P

l

P

l

n

b

+ =

=

å

=

-( )

2 2 1

,

0, N 1,

N j n n j j j

b

h l

P

l

n

b

=

å

=

-h

r h a

_n

= =

m

or

n M

>

( ) ( )

1 1

,

0,

1,

N j n n j n j j j

a

l

P

l

P

l

n

M

b

+ =

=

å

=

-( )

1

( )

1

( )

1 1

,

N N j j M M j M j M M j M j j j j j

a

al

P

l

P

l

c

l

P

l

P

l

b

+

b

+ = =

=

å

=

å

( ) ( )

1 1

,

1,

2,

N j n n j n j j j

c

al

P

l

P

l

n M

N

b

+ =

=

å

=

+

-( )

2 1

,

0,

,

N j n n j j j

b

l

P

l

n

M

b

=

å

=

( )

2 1

,

1,

1.

N j n n j j j

d

al

P

l

n M

N

b

=

å

=

+

-J

{

l b

_k

,

_k

}

{

l b

_k

,

_k

}

J J k j

l

¹

l

1

N j=

b

j

=

å

a

_n

>

0,

n

=

0,

N

-

2

(8)

Lemma 6. Let are distinct real numbers and for the positive numbers be given that . Then there exists unique polynomials

with and positive leading coefficients satisfying the conditions (11).

Now, we will give another method for a kind of approach to the solution of the inverse spectral problem which is called the Gelphand-Levitan-Marchenko method.

Let be a solution of the Eqn. (1) satisfying the conditions

,

in the case .

Recall that is a polynomial of degree , so it can be expressed as

, (19)

where and are constants. There is a connection between coefficients and .

Then we can write the equalities

(20)

(21)

Now, we can write from Eqn. (19)

,

1,

k

N

l

=

,

1,

k

N

b

=

1

N j=

b

j

=

å

( )

,

0,

1

k

P

l

k

=

N

-

deg

P

_j

( )

l =

j

( )

n

R

l

( )

1

0,

0

1 R

_-

l

=

R

l

=

1,

0

n n

a

º

b

º

( )

n

P

l

n

( )

1 ,

( )

{

}

0

,

0,1,..., ,...,

n n n n n k k k

P

l

g

R

l

-

c

R

l

n

M

N

=

æ

ö

=

_ç

+

_÷

Î

è

å

ø

n

g

c

_{n k}_,

a b c d

_n

, , ,

_n _n _n ,

,

n n k

g c

(

)

(

)

0 1 1 1

0 ,

1,

2 ,

,

n n n n M n M n M

a

n M

c

M

n N

c

g

_g

g

ag

+ + +

=

£ £

=

< £ -

=

(

)

(

)

, 1 1, 0, 1 , 1 1,

0 ,

0,

1 .

n n n n n n n n n n

b

n M

d

M n N

c

- + -- +

=

-

£ £

=

-

< £

(9)

-, (22)

where is defined by (12) and

. (23)

Since the expansion

holds, then from Eqn. (11) we have

.

Considering the Eqn. (22) we get

(24)

(25)

Eqn. (24) is important for the solution of inverse spectral problem. Firstly, are determined by using Eqn. (23) and then quantities are found from system of Eqn. (24). Thus we can find unknowns with aid of from Eqn. (25).

Lemma 7. For any fixed the system of Eqn. (24) is identically solvable. Proof. It is clear that

,

are linear independent and from the Eqn. (23), we have

( ) ( )

1 , 1 0 N n n j m j n nm n k km j j k

P

R

G

h

_l

_g

_c

b

-= =

é

ù

=

_ê

+

_ú

ë

û

å

h

( ) ( )

1 N nm n j m j j j

G

h

R

l

R

l

b

=

å

( )

(j)

( )

0 j j k k k R

l

w P

l

= =

å

( ) ( )

1

1 ,

0,

N n j m j nm j j n

P

l

R

l

d

n

s

n

b

hg

=

³

=

å

1 , 0 0, 0, 1, 1, n nm n k km k G -

c

G m n n = +

å

= = - ³ 1 , 2 0

1 ,

0, N 1.

n nn n k kn k n

G

c

G

n

hg

-=

+

å

=

-nm

G

,

0,

1

n k

k

n

c

=

-n

g

c

_{n k}_,

n

( )

1

( )

2

( )

1 2

,

,...,

,

0,

1

n n n N n N

R

v

l

n

N

b

æ

ö

=

ç

_ç

÷

_÷

=

-è

ø

(10)

.

The basic determinant of the system (24)

. (26)

Lemma 8. Let be a solution of the system (24). Then

Proof. See [19].

Therefore we can determine the entries of the matrix from the formulas Eqn. (20) and Eqn. (21).

4. Inverse Problem for Two Spectra

Consider the boundary value problem

(27) with the boundary conditions

, (28)

where is defined in (3). Now, the matrix of coefficients of Eqn. (27) is denoted by which has the same form with matrix . If we delete the first row and the first column of the matrix then we have tridiagonal matrix which has number eigenvalues

. Assume that eigenvalues of matrix are distinct and real. Thus we can write

.

The solution of Eqn. (27) is denoted by provided that . It is clear that the eigenvalues are zeros of the polynomial . While we determine entries of , we will use eigenvalues of matrices and .

(

,

)

ns n s

G

=

v v

{ }

1

{

(

)

}

1 , 0 , 0

det

_ij n

det

_i

,

_j n i j i j

G

-₌

v v

-=

=

,

0,

1

nm

G

m

=

n

-1 , 0 0, 0, N 1. n nm n k km k G -

c

G n = +

å

> = -J 1 1 1

,

1,

n n n n n n n n

a y

_- _-

+

b y

+

a y

₊

=

lr

y

n

=

N

-0 N

0 y

=

y

=

n

r

J

₁ J J 1 1 N- ´ -N

J

₁ N-1

,

1,

1

k

N

µ

=

-

J

1 1 2

...

N 1

µ µ

<

< <

µ

-( )

{

Q

n

l

}

Q

0

( )

l

=

0,

Q

1

( )

l

=

1 ,

1,

1

j

N

µ

=

-

Q

_N

( )

l

J J

J

₁

(11)

Now we will give an important lemma for the inverse spectral problem according to the two spectra.

Lemma 9. The eigenvalues of matrices and alternate, i.e.

.

Proof. See [19].

Additionally, we can find the normalized numbers by aid of two spectrums and of the matrices and respectively. Assume that

, (29)

and require that . is a meromorphic function,

, (30)

its poles and zeros coincide with the eigenvalues of the problem (1)-(2) and (27)-(28), respectively. We see that the function satifies the equation

(31)

Now, if the equality Eqn. (31) is multiplied by and the Eqn. (6) (for ) is multiplied by then the second result is substracted from the first and sum by , we obtain:

or

,

and then for , we have

J

₁ 1 1 2 2

...

N 1 N 1 N

l µ l

<

µ

< <

l

-

<

µ

-

<

l

k

b

1

, ,...,

2 N

l l

l

µ µ

₁

, ,...,

₂

µ

_N-₁ J

J

1

( )

( ) ( ) ( )

n n n

f

l

=

Q

l

+

m

l

P

l

( )

0

N

f

l =

m

( )

l

( )

N

_{( )}

( )

N

Q

m

P

l

=

-( )

n

f

l

( )

1 1 1

.

n n n n n n n n

a f

_- _-

l

+

b f

l

+

a f

₊

l lr

=

f

l

( )

n k

P

l

l l

=

_k

( )

n

f

l

n

(

)

( ) ( )

(

( ) ( )

)

( )

(

)

1 1 ₁ ₁ ₁ 1 1 ₁ ₁ N N _n _n _n _k _n _k _n k n n n k n n _n _n _n _k _n _k _n

a

f

P

f

P

a f

P

f

l

l l

r

l

- - _- _- -= = ₊ ₊

ì

-

ü

ï

-

_{= í}

_ý

-

-ï

ï

î

þ

å

(

)

1

( ) ( )

0 1 N k n n n k n f P a

l l

-

r

l

= -

å

= -k

l

®

l

(12)

.

On the other hand from Lemma 1, we get

,

and from these equalities we can find

.

As a result

. (32)

Theorem 10. Let the collections to be real numbers. These collections are spectrums of the and tridiagonal almost-symmetric matrices and , respectively, it is necessary and sufficient that they are alternate as below:

.

5. Conclusion

While solving the inverse spectral problem from two spectra, firstly we determine the normalized numbers of the matrix by using the eigenvalues of the matrices and , respectively. Thus, we reduce the problem from two spectra to spectral analysis, and then we determine the values , and from the formulas (23)-(25) by aid of the eigenvalues and the normalized numbers of the matrix .

( )

0 N k k N k

a P

Q

l

b

l

¢

=

(

) ( )

0 1 1 1

( )

det

J

-

l

I

= -

1

N

a a a c

...

_{M M}₊

...

c P

_N_- _N

l

(

) ( )

1

( )

1 1 1 1

det

J

-

l

I

= -

1

N-

a a c

...

_{M M}+

...

c Q

_N- _N

l

( ) (

1

) (

)

( ) (

1

) (

1

)

0 1 1 1 1 1 1

...

,

...

N N N N M M N M M N

P

Q

a a a c

c

a a c

c

l l

l µ

l

-+ - +

--

-

-=

=

(

)

(

)

1 1 1 N k j j j k k N k j j

l l

b

l

µ

= ¹ -=

-=

-Õ

Õ

{ }

1 1

,

1 N N k k k k

l

µ

-= = N N´ N- ´ -1 N 1 J

J

₁ 1 1 2 2

...

N 1 N 1 N

l µ l

<

µ

< <

l

_-

<

µ

_-

<

l

k

b

J

{ }

1 1

,

1 N N k _k k _k

l

µ

-= = J

J

₁ nm

G

c

_n_,k

g

n k

l

b

_k J

(13)

Consequently, the entries and are found from the Eqn. (20) and Eqn. (21). Thus, the matrix is reconstructed by using Parseval formula.

References

[1] Hochstadt, H., On some inverse problems in matrix theory, Archiv der Mathematik, 18(2), 201-207, 1967.

[2] Hochstadt, H., On the construction of a Jacobi matrix from spectral data, Linear Algebra and Its Applications, 8(5), 435-446, 1974.

[3] Hochstadt, H., On the construction of a Jacobi matrix from mixed given data, Linear Algebra and its Applications, 28, 113-115, 1979.

[4] Fu, L., Hochstadt, H., Inverse theorems for Jacobi matrices, Journal of Mathematical Analysis and Applications, 47(1), 162-168, 1974.

[5] Gasymov, M.G., Guseinov, G.Sh., On inverse problems of spectral analysis for infinite

Jacobi matrices in the limit-circle case, In Soviet Mathematics Doklady, 40, 627-630, 1990.

[6] Guseinov, G.Sh., Determination of an infinite non-selfadjoint Jacobi matrix from its

generalized spectral function, Matematicheskie Zametki, 23, 237-248, 1978; English transl.:

Mathematical Notes, 23, 130-136, 1978.

[7] Guseinov, G.Sh., The inverse problem from the generalized spectral matrix for a second

order non-selfadjoint difference equation on the axis, Izvetsiya Akademii Nauk Azerb. SSR Ser.

Fiz.-Tekhn. Mat. Nauk, 5, 16-22, 1978 (in Russian)

[8] Guseinov, G.Sh., An inverse spectral problem for complex Jacobi matrices, Communications in Nonlinear Science and Numerical Simulation, 15(4), 840-851, 2010.

[9] Guseinov, G.Sh., Construction of a complex Jacobi matrix from two-spectra, Hacettepe Journal of Mathematics and Statistics, 40(2), 2011.

[10] Guseinov, G. Sh., On construction of a complex finite Jacobi matrix from two-spectra, Electronic Journal of Linear Algebra, 26, 101-120, 2013.

[11] Guseinov, G.Sh., On a discrete inverse problem for two spectra, Discrete Dynamics in Nature and Society, 14 pages, 2012.

[12] Guseinov, G.Sh., On an inverse problem for two spectra of finite Jacobi matrices, Applied Mathematics and Computation, 218(14), 7573-7589, 2012.

[13] Guseinov, G.Sh., Inverse spectral problems for tridiagonal N by N complex

hamiltonians, Symmetry Integrability and Geometry: Methods and Applications, 28 pages, 2009.

[14] Manafov, M.D., Bala, B., Inverse spectral problems for tridiagonal N by N complex

hamiltonians with spectral parameter in the initial conditions, Adıyaman Üniversitesi Fen

Bilimleri Dergisi, 3(1), 20-27, 2013.

[15] Akhmedova, E.N., Huseynov, H.M., On eigenvalues and eigenfunctions of one class

of Sturm-Liouville operators with discontinuous coefficients, Transactions of National Academy

of Sciences of Azerbaijan Series of Physical-Technical and Mathematical Sciences, 23-4, 7-18, 2003.

[16] Akhmedova, E.N., Huseynov, H.M., On inverse problem for Sturm-Liouville

operators with discontinuous coefficients, Transactions of Saratov University, (Izv. Sarat. Univ.),

10(1), 3-9, 2010.

, ,

n n n

a b c

d

_n

(14)

[17] Bala, B., Kablan, A., Manafov, M.Dzh., Direct and inverse spectral problems for

discrete Sturm-Liouville problem with generalized function potential, Advances in Difference

Equations, 2016(1), 172, 2016.

[18] Huseynov, H.M., Finite dimensional inverse problems, Transactions of National Academy of Sciences of Azerbaijan Series of Physical-Technical and Mathematical Sciences, 21(1), 80-87, 2001.

[19] Bala, B., Manafov, M.Dzh., Kablan, A., Inverse spectral problems for spectral data

and two spectra of N by N tridiagonal almost-symmetric matrices, Applications and Applied

Mathematics, 14(2), 12 pages, 2019.

[20] Manafov, M.Dzh., Kablan, A., Bala, B., Parseval equality of discrete Sturm-Liouville

equation with periodic generalized function potentials, AIP Conference Proceedings. Vol. 1991.

No. 1. AIP Publishing, 2018.

[21] Baş, E., Özarslan, R., A new approach for higher-order difference equations and

eigenvalue problems via physical potentials, The European Physical Journal Plus, 134(6), 253,